ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbi1v Unicode version

Theorem sbi1v 1771
Description: Forward direction of sbimv 1773. (Contributed by Jim Kingdon, 25-Dec-2017.)
Assertion
Ref Expression
sbi1v  |-  ( [ y  /  x ]
( ph  ->  ps )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem sbi1v
StepHypRef Expression
1 sb6 1766 . 2  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
2 sb6 1766 . . 3  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  A. x ( x  =  y  ->  ( ph  ->  ps ) ) )
3 ax-2 6 . . . . 5  |-  ( ( x  =  y  -> 
( ph  ->  ps )
)  ->  ( (
x  =  y  ->  ph )  ->  ( x  =  y  ->  ps ) ) )
43al2imi 1347 . . . 4  |-  ( A. x ( x  =  y  ->  ( ph  ->  ps ) )  -> 
( A. x ( x  =  y  ->  ph )  ->  A. x
( x  =  y  ->  ps ) ) )
5 sb2 1650 . . . 4  |-  ( A. x ( x  =  y  ->  ps )  ->  [ y  /  x ] ps )
64, 5syl6 29 . . 3  |-  ( A. x ( x  =  y  ->  ( ph  ->  ps ) )  -> 
( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ps ) )
72, 6sylbi 114 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  ->  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ps ) )
81, 7syl5bi 141 1  |-  ( [ y  /  x ]
( ph  ->  ps )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1241   [wsb 1645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427
This theorem depends on definitions:  df-bi 110  df-sb 1646
This theorem is referenced by:  sbimv  1773
  Copyright terms: Public domain W3C validator